# Calculation of psi when reference exon has a junction with the next exon (0 exon excluded)

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Assigned to pje...@biociphers.org by me

### Elsa Claude

Mar 1, 2021, 12:21:15 PMMar 1
to majiq_voila
Hi,

Thanks again for such a useful tool.
I have a question regarding the values of psi in an lsv. Thanks to the article you wrote, I understand that the sum of all psi in a LSV must be equal to 1. In general, psi calculation is : #includingreads / (#includingreads + #excludingreads), if I remember well.
So how is it possible that, for example, in a source based LSV such as s|1e1.1o1|1e2.1o1 , that the first junction (which is basically the reference exon being linked to the next exon, without excluding any exon) has a value different from 1 ? Because in this case, the ratio used to calculate psi value, should be for example : 4/4.
Does MAJIQ calculate the psi value of this junction by doing : 1 - (sum(all other junctions in the LSV)) ?

The answer is maybe is the article but I am struggling a bit with the mathmatical formulas. :/

Any help would be highly appreciated !
Have a nice day,
Cheers,

Elsa

### Paul Jewell

Mar 4, 2021, 12:16:47 PMMar 4
to majiq_voila
Hi Elsa,

If I'm understanding your question correctly, the calculation is not quite as simple as you describe. However, there should not be large discrepancies in a single lsv, where the total is not ~equal to 1.

Can you please post a screenshot of the lsv and values you see for it's junctions, so that I of one of the lab members may be able to easily spot what is happening?

Thanks,
-Paul

### Elsa Claude

Mar 5, 2021, 8:46:21 AMMar 5
to majiq_voila
Yes apparently the calculation is not that simple... I supposed that the part of the paper that I am interesting in is :

Estimating the percent selected index (PSI, or Ψ) per junction j in a given LSV e in experiment t requires to derive a posterior distribution over multinomial distributions Ψe = {Ψe,j}J, s.t. ∑ Ψe,j = 1, ∀e,j 0 ≤ Ψe,j ≤ 1. Previous works concentrated on common cases involving two junctions such as cassette exons, where the posterior for Ψ can be computed in closed form using for example a Beta prior. For more complex cases where J > 2 the PSI posterior was commonly computed either as a point estimate (e.g., ML estimator using EM) or using MCMC sampling techniques (Katz et al., 2010). In general, sampling based estimation for Ψe,j or ∆Ψe,j (below) scales exponentially in the number of junctions J and is also hard to visualize beyond J = 2. However, noting that in most cases researchers are interested in relative abundance of specific variants rather than a complete distribution over all isoforms, MAJIQ side steps these issues by computing only the posterior marginal distributions per variant. This computation scales linearly with J and simplifies both downstream analysis and visualization of the results. It has been previously observed that alternative junctions in a given experimental condition generally tend to be either highly included or highly excluded (Shen et al., 2012; Wu et al., 2011). In line with these observations and based on fitting empirical distributions of observed PSI (data not shown) MAJIQ uses the following prior: P0(Ψe,j)~Beta(α=1Jη,β=J1Jη). The default is η=1 resulting in a Jeffery prior that encourages either high inclusion or exclusion levels, but any (α, β) can be set. MAJIQ then uses the M read samples per junction (see above) to get a posterior Ψe,j as an average over those posterior distributions.

I struggle a little to apply the formula to a practical case. Could you help me for example to understand what would be the calculation of psi for example in this LSV, for the red junction ?

Thank you for the help !

Have a nice day :)

Elsa

### Paul Jewell

Mar 5, 2021, 11:35:32 AMMar 5
to majiq_voila
Hi Elsa,

I'm going to reach out to some of my colleagues for help with decoding the theoretical question you have, please stand by for that reply.

I also wanted to ask, though, if there was indeed a discrepancy here, with regard to the three psi values not adding to ~1? Could you post the three psi values for grp1 and/or grp2? They are not coming out to be approximately 1, it may be something else I need to investigate.

Thanks,
-Paul

### Elsa Claude

Mar 5, 2021, 1:47:53 PMMar 5
to majiq_voila
Oh sorry that's my fault, I gave a deltapsi example, was not paying attention.
Here is an example with a psi voila view.
I am interesting on the calculation used to find psi value of the red junction.